Dark sector may couple to the Standard Model via one or more mediator particles. We discuss two types of mediators: the dark photon $A^{prime}$ and the dark scalar mediator $phi$. The total cross-sections and various differential distributions of the processes $e^{+} e^{-} rightarrow q bar{q} A^{prime}$ and $e^{+} e^{-} rightarrow q bar{q} phi$ ($q=u,~d,~c,~s$ and $b$ quarks) are discussed. We focus on the study of the invisible $A^{prime}$ due to the cleaner background at future $e^{+} e^{-}$ colliders. It is found that the kinematic distributions of the two-jet system could be used to identify (or exclude) the dark photon and the dark scalar mediator, as well as to distinguish between them. We further study the possibility of a search for dark photons at a future CEPC experiment with $sqrt{s}=$ 91.2 GeV and 240 GeV. With CEPC running at $sqrt{s}=$ 91.2 GeV, it would be possible to perform a decisive measurement of the dark photon (20 GeV $< m_{A^{prime}} <$ 60 GeV) in less than one operating year. The lower limits of the integrated luminosity for the significance $S/sqrt{B}=$ 2$sigma$, 3$sigma$ and 5$sigma$ are presented.
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